The product of the integers from $1$ through $7$ is equal to $2^j\cdot3^k\cdot 5 \cdot 7$. What is the value of $j-k$?
Explanation: The product of the integers $1$ through $7$ is $1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7$. We can re-write some of these numbers, however, as $4 = 2\cdot2$ and $6 = 2\cdot 3$, and so $1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7 = 2\cdot 3\cdot 2\cdot 2\cdot 5\cdot 2\cdot 3\cdot 7 = 2^4\cdot 3^2\cdot 5\cdot 7$. Thus, $j = 4$ and $k = 2$, so $j-k = 4-2 = \boxed{2}$.